Symmetry breaking in Grasshopper-type Ising models

Distance-selective interactions in Ising systems give rise to a range of rich and unexpected phenomena. A notable example is the so-called Grasshopper Ising model, a conserved spin Ising model with interactions only between spins at a fixed distance. This model is the discrete version of the following mathematical problem: A grasshopper lands at a random point on a planar lawn of area one. It then makes one jump of fixed distance in a random direction. What lawn shape maximizes the probability that the grasshopper lands on the lawn after the jump? The solution to this problem, corresponding to the ground state of the Grasshopper Ising model, exhibits spontaneous breaking of rotational symmetry and the emergence of cogwheel-like structures and other unexpected shapes, which also resemble patterns reported in simulations of dipolar Bose-Einstein condensates.

In this poster, we explore these symmetry breaking properties in several experimentally relevant variants of the model. To accurately approximate the continuous grasshopper problem, the fixed interaction range of the corresponding Ising model must be large. However, potential experimental realizations (for instance with Rydberg atom arrays) are likely limited to small interaction ranges. With this in mind, we analyze the Grasshopper Ising model with small numbers of spins, which correspond to smaller interaction ranges. We identify the minimum system size required for the emergence of recognizable cogwheel structures and reveal a power-law dependence of cogwheel height on system size. We also discuss a variant of the model with smeared interactions that have a finite Gaussian width and identify critical widths beyond which rotational symmetry is restored.